1 [30 points] Mark each of the following as true or false and for those that are

Question

1 [30 points] Mark each of the following as true or false and for those that are false rewrite the statement on the space below the question so as to make it a true statement. (a) TrueIf two matrices are row equivalent to each other then their row spaces are the same. (b) ___ _ If the nullity of a 7 × 8 matrix is 4, then the column space is of dimension 4 as well. (c)A vector u is in the column space of matrix A if and only u=Ax for some vector x. If a set of n vectors in R" is linearly independent, then the set is a spanning set. The coordinate vector forthe vector u = (1,5,1) rel- 2 ative to the basis 3 4is (1, 1, 1). The rank of a matrix is always equal to the dimension of its null space. If a vector space has dimension k, then any set of more than k vectors in that space must be linearly independent. (h) If an rn × n matrix A has rank n, then m

Explanation / Answer

(a) true

(b) true

(c) true

(d) true

(e) true

(f) false

The rank of a matrix is always equal to, r = n - (dimension of null space) , where n is the number of columns of the matrix

(g) false

If a vector space has dimension k, then any set of more than k vectors in that space must be linearly dependent.

(h) false

If an m*n matrix A has rank n, then, n <= m

(i) true

(j) true

If you want justification of each answers, let me know in the comment section, I will come up with the solution at the earliest.

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